Quasi-classical Ground States. II. Standard Model of Non-relativistic QED

TL;DR

This paper analyzes the ground states of a non-relativistic electron coupled with the quantized electromagnetic field, establishing existence results without cutoffs and deriving asymptotic energy expansions.

Contribution

It proves the existence of ground states in non-relativistic QED without ultraviolet or infrared cutoffs and derives asymptotic expansions of the ground state energy.

Findings

Existence of ground states under general conditions

No need for ultraviolet or infrared cutoffs

Second-order asymptotic expansion of ground state energy

Abstract

We consider a non-relativistic electron bound by an external potential and coupled to the quantized electromagnetic field in the standard model of non-relativistic QED. We compute the energy functional of product states of the form $u\otimes \Psi_f$, where $u$ is a normalized state for the electron and $\Psi_f$ is a coherent state in Fock space for the photon field. The minimization of this functional yields a Maxwell--Schr{\"o}dinger system up to a trivial renormalization. We prove the existence of a ground state under general conditions on the external potential and the coupling. In particular, neither an ultraviolet cutoff nor an infrared cutoff needs to be imposed. Our results provide the convergence in the ultraviolet limit and the second-order asymptotic expansion in the coupling constant of the ground state energy of Maxwell--Schr\"odinger systems.